Gerald Williams

The aspherical Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups

Abstract The Cavicchioli–Hegenbarth–Repovš generalized Fibonacci groups are defined by the presentations Gn (m, k) = 〈x 1, … , xn | xixi+m = xi+k (1 ⩽ i ⩽ n)〉. These cyclically presented groups generalize Conways Fibonacci groups and the Sieradski groups. Building on a theorem of Bardakov and Vesnin we classify the aspherical presentations Gn (m, k). We determine when Gn (m, k) has infinite abelianization and provide sufficient conditions for Gn (m, k) to be perfect. We conjecture that these are also necessary conditions. Combined with our asphericity theorem, a proof of this conjecture would imply a classification of the finite Cavicchioli–Hegenbarth–Repovš groups.

UNIMODULAR INTEGER CIRCULANTS ASSOCIATED WITH TRINOMIALS

The n × n circulant matrix associated with the polynomial (with d < n) is the one with first row (a0 ⋯ ad 0 ⋯ 0). The problem as to when such circulants are unimodular arises in the theory of cyclically presented groups and leads to the following question, previously studied by Odoni and Cremona: when is Res(f(t), tn-1) = ±1? We give a complete answer to this question for trinomials f(t) = tm ± tk ± 1. Our main result was conjectured by the author in an earlier paper and (with two exceptions) implies the classification of the finite Cavicchioli–Hegenbarth–Repovs generalized Fibonacci groups, thus giving an almost complete answer to a question of Bardakov and Vesnin.

Efficient finite groups arising in the study of relative asphericity

We study a class of two-generator two-relator groups, denoted

Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups

J_n(m,k)

On the cohomology of Generalized Triangle Groups

Jn(m,k), that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature as finite groups of intriguing orders. Here we find infinite families of non-elementary virtually free groups and of finite metabelian non-nilpotent groups, for which we determine the orders. All Mersenne primes arise as factors of the orders of the non-metacyclic groups in the class, as do all primes from other conjecturally infinite families of primes. We classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups

LARGENESS AND SQ-UNIVERSALITY OF CYCLICALLY PRESENTED GROUPS

F^{a,b,c}